Factor fully then graph the following function.

Question

HELP ME!!!! I need to factor this using synthetic division and factoring and stuff but have no idea how!

 

24x^8 - 194x^7 + 201x^6 + 1713x^5 - 4203x^4 + 657x^3 + 4458x^2 - 2176x - 480

Answer 1

A trick before you do synthetic division:  plot the function or simply chart the value at integers to find the zeros.  Using the rational root theorem with 480 and 24 is a possibility, but it will take longer.

Actually, once seeing the answer, we see that the plot should be from -4 to 5.

We quickly see that -3, -1, 1, 2, and 4 are roots.  Let's do synthetic division with this.

I will do this in Excel to speed the process.

 Once I had the cubic, I recharted and saw that 4 was again a root.  We can also tell this because the function has the same sign on both sides of 4.

This left me with a quadratic.  24x^2 - 26x - 5.

I could then rechart, use the quadratic formula, or factor.

The charting worked so well, let's keep this up.

I see that 5/4 is a root.

The final root can be determined by completing the synthetic division.

We get 24x + 4 = 0, so x = -1/6 is the final root.

Roots are -3, -1, -1/6, 1, 5/4, 2, 4, and 4

 

The completed synthetic division is below but is quite ugly.

 

 

-3 24 -194 201 1713 -4203 657 4458 -2176 -480
          -72 798 -2997 3852 1053 -5130 2016 480
-1 24 -266 999 -1284 -351 1710 -672 -160      0
           -24 290 -1289 2573 -2222 512 160
  1 24 -290 1289 -2573 2222 -512 -160   0
             24 -266 1023 -1550  672   160
  2 24  -266 1023 -1550 672   160      0
              48 -436 1174 -752 -160
  4 24 -218 587 -376 -80 0
             96 -488 396 80
  4 24 -122    99   20   0
             96 -104 -20
1.25 24 -26    -5    0
              30     5
-1/6  24    4     0
               -4
         24    0